Definition:Inductive Semigroup
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Definition
Let $\struct {S, \circ}$ be a semigroup.
Let there exist $\alpha, \beta \in S$ such that the only subset of $S$ containing both $\alpha$ and $x \circ \beta$ whenever it contains $x$ is $S$ itself.
That is:
- $\exists \alpha, \beta \in S: \forall A \subseteq S: \paren {\alpha \in A \land \paren {\forall x \in A: x \circ \beta \in A} } \implies A = S$
Then $\struct {S, \circ}$ is an inductive semigroup.
Also see
- Results about inductive semigroups can be found here.
Sources
- 1960: Seth Warner: Mathematical Induction in Commutative Semigroups (Amer. Math. Monthly Vol. 67, no. 6: pp. 533 – 537) www.jstor.org/stable/2309170
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Exercise $16.9$